(For a reasonably convincing argument on why the letter τ (tau) in particular should be adopted for this value, please read Mr. Hartl's manifesto.)

The fundamental point here is that, in trigonometry and all other manner of angle-measuring endeavors, what we care about is the radius of a circle, not its diameter. The one follows from the other to be sure, but at the end of the day the diameter is more usefully considered twice the radius than the radius is half the diameter. A circle is a circumference around a center—it is the measure of this distance between center and circumference that is elemental to the idea of a circle, not the rather incidental fact that its full width is twice that same distance.

To be sure, although 2π features prominently in all areas of mathematics dealing with angles and arc, regular π comes up a lot too. But dealing with τ in these situations would never be more of an issue than simply dividing τ by two or what have you. Again, the important point is that τ should properly be considered the more fundamental value, so even in cases were π leads to less verbose or more eloquent expressions, it would still perhaps be more informative and pertinent to use τ. Though perhaps not in all cases, I don't know. At any rate, there is no need for π to be "abolished" or anything, in the way that, say adopting dozenal notation would probably require the death—in practical usage—of the decimal radix. Both values can go on to have long and fulfilling lives. τ can be propagated in common usage by simply defining it and using it whenever π is required—there is no need for any sort of radical, society-wide transformation. Indeed, I can easily envision a situation where τ (or some other symbol, if τ ends up being impractical) slowly overtakes π as a the preeminent circle constant over the course of decades if not centuries.

The argument was presented so well by Mr. Palais and so exhaustively by Mr. Hartl that there scarcely much more for me to add on the matter—except perhaps to note that, of course, the prominence of six in the value of τ definitely highlights the hexagonal nature of the circle. Every time one considers the value of τ one is reminded that the circle differs from the hexagon by only 0;3494 radians of perimeter. Which is, in this blog's opinion, a worthwhile thing to remember.

Anyway, it is my intention going forward to use τ instead of π in all applicable formulae, expressions, equations, and what have you throughout this site. I'm not sure how well I'll stick to that plan, but this certainly seems like an easier transition to make than the whole dozenal enterprise would be. One can always simply define τ on the fly for those who are unfamiliar with it, while it is very difficult to just go right out and use dozenal notation without at least explaining yourself first. But I consider both of these efforts to be part of the same grand movement to bring clarity and hexagonal sensibilities to modern mathematical semantics.

I am not yet sure what the ramifications of this development will be for Dozenal Pi Day (March 18;), since there is conspicuously no 34th day in June—in decimal or dozenal. Mr. Hartl apparently celebrates decimal "Tau Day" on June 28th (a perfect day in decimal), but my own preference would be to have a "universal" Tau Day on February 27., being 365./τ, though that is probably a bit too cerebral for most people, and it sacrifices the dozenal element. At any rate, I shall move forward with plans to celebrate Dozenal Pi Day in 11B7;—it is still τ/2 after all—and if nothing else I can use it as an opportunity to denounce both decimalism and π.

Wrong way.

https://plus.google.com/b/115025073359073355980/108416959794948591172/posts/ZvwDGyfUWwK
spaceLem commented on a video on YouTube.
Shared publicly - Mar 7, 2015
I'm a base 12 supporter! (And a tau supporter).
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spaceLem
Mar 23, 2015+
+scaper8 About the only time pi is simpler than tau is when you're measuring pipes, because it's easier to measure the diameter than the radius. For pretty much all other uses, tau is easier, because you inevitably end up needing to use 2*pi. What's that? The area of the circle is easier with pi (pi*r^2 vs 1/2 tau r*2)? Well actually no, because the circle is an area, and most areas are integrals and have a half in them (consider 1/2 mv^2, 1/2*base*height, and other examples that I can't remember off the top of my head). But mainly it's because tau converts easily to "turn", and that's so much easier than working in half turns, which pi gives you.

scaper8
Mar 23, 2015
Maybe that's part of the problem with me. I just don't think in terms of "turns." That argument seems useless from the way my mind works; and since I'm already used to pi... I guess if I were introduced to it early on instead of pi, maybe then. :-)

alysdexia
Mar 29, 2015
+spaceLem I support base-12, but not under those silly nonEnglish names the Dozenal Society use. However I am wont to pi/2. I think I'll call it ýpsilon for hýpoteinusa. Its numerical coefficient in expressions may be 2, 4, 8, etc. dependent on dimensionality.

Cooper Gates
Apr 1, 2015
+scaper8 +spaceLem One turn = one revolution = one circumference movement of edge (circle's version of perimeter). However, think of the intercepts and asymptotes of trig functions: n * pi for sine and (1/2 + n) * pi for cosine and tangent, etc. etc.; those become n * tau / 2 and (1/2 + n) * tau / 2; that second one may also be written tau / 4 + n * tau / 2.

spaceLem
Apr 2, 2015
Well the intercepts correspond to where revolutions cross the x-axis. e^i tau/2 = -1 means a half rotation takes you to the other side of the x-axis, e^i tau = 1 means a full revolution returns you to where you were. As for the unfortunate fractions, they still work, they just correspond to that fraction of a turn.

spaceLem
Apr 24, 2015
+IAmAPerson Because a) it matches formulae for other area integrals (1/2 x^2 is very common), and b) it's just about the only time when it simplifies a formula. Do we have to put up with a complication in every other formula to make this one a tad easier?

IAmAPerson
Apr 24, 2015
+spaceLem "it matches formulae for other area integrals"
When in math do we NOT simplify something? Essentially never.
"it's just about the only time when it simplifies a formula"
F=(π²EI)/L² versus F=(τ²EI)/2L²
Integral from -∞ to ∞ of the hyperbolic secant of x is exactly π. Not τ
e^iπ+1=0 where e^iτ-1=0 has an ugly -1 in it.
Other formulae, like SA=4πr² aren't made any better by τ, and is just a pain to adjust everything.

IAmAPerson
Apr 24, 2015
Also, WHY NOT JUST USE BOTH? Why must we choose pi or tau? Use whichever is easiest!

Cooper Gates
Apr 24, 2015
+IAmAPerson Yep it depends on the formula, just look for places where you find a 2Pi and you can put taus in there (or a 4Pi^2 = tau^2)

spaceLem
Apr 24, 2015
+IAmAPerson Apparently not often enough, because 2pi occurs all over the place: the normal distribution, the Fourier transform (I use those two quite a bit myself), the roots of unity.

And those constants are there to aid in understanding of what's going on: 1/2 tau R^2 is just a special case of the integral 1/2 theta R^2.

If the integral of sech(x) is 1/2 tau, there's probably a geometric reason for that. pi makes all the trig formulae unnecessarily complicated to follow geometrically: sin of 1/8 of a turn, vs sin of 1/4 of a half turn... really? and e^i tau =1 is perfect: one rotation gets you back where you started. e^i tau/2 = -1 is perfect: one half rotation gets you the negative of what you had. Tau makes that just intuitive. People are only surprised by that result because pi is so unintuitive.

IAmAPerson
Apr 24, 2015
+spaceLem You continue to dodge my question on why we can't have pi and tau in active use.

spaceLem
Apr 24, 2015
+IAmAPerson Sorry, I hadn't even noticed that question (and technically if I had, I only dodged it once, I didn't continue to dodge it). Use pi if you must (and let's face it, you've got momentum with you), but I'm convinced that tau is the better constant, and continuing to use pi 1) won't help people to let it go, and 2) I don't really think having two similar constants is really helpful, especially when one of which is just defined as half of the other?

spaceLem
Apr 25, 2015+
+alysdexia It's not arbitrary. The normalising constant is because to integrate exp(x^2) involves changing coordinates and integrating around a circle at some point (or something -- it's been years since I actually did this in maths).

alysdexia
Apr 25, 2015
+spaceLem
Well the exponent is arbitrary too.

spaceLem
Apr 25, 2015+
+alysdexia No it really isn't. The normal distribution is a very special distribution because of the central limit theorem, which says that the sum (and mean) of multiple iid values with identical mean and variance tends to the normal distribution as the number of samples increases, regardless of the underlying distribution. The exponent is fixed.

alysdexia
Apr 26, 2015
+spaceLem In real life terms are not wholly independent (perturbation), linearly-additive (relativity), or continvose (atomism) and conditions are not static (nonconservative field) or closed (nonequilibrium).

spaceLem
Apr 26, 2015
+alysdexia I really don't follow what you meant there. How is any of that relevant to the geometric significance of tau in the normal distribution?

alysdexia
Apr 26, 2015
+spaceLem
If you presuppose unitarity and two dimensions, you'll get τ. However I am more interested in the fundamental or elementary υ = τ/4.

spaceLem
Apr 26, 2015+
+alysdexia A quarter turn? Useful for when things are perpendicular, but again I'm not seeing the relevance. I also don't recall encountering it (I could be wrong, feel free to remind me what it's for), so I'll suggest that it's not a very important value to start life knowing about. Edit: I went back through the conversation thread and saw you bring it up ages ago, although I don't think you mentioned it again until now, and haven't really explained any situations when it's useful.

alysdexia
Apr 30, 2015
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+spaceLem It's common courtesy to reduce all constants to their LCF.

spaceLem
Apr 30, 2015
+alysdexia I'm not sure if I particularly want to continue this, or if I'm going to receive more incoherent nonsense if I do.

What is the formula for the area of a slice of a circle? 1/2 theta r^2. So when theta = the whole circle (tau radians) you get 1/2 tau r^2. If you replace tau with pi, you get 1/2 (2 pi) r, which simplifies to pi r^2. This formula appears easier, but factoring out the constant has just made the general case harder to see.

So no, you don't always want to reduce constants, not when you're obscuring what's actually happening by doing so.

alysdexia
Apr 30, 2015
+spaceLem I gave you no incoherent nonsense. Your argument is grounded on irrelevant form. Leave the coefficient however you want. You don't start counting at a base and work backwards towards the unit.

spaceLem
Apr 30, 2015
+alysdexia "I gave you no incoherent nonsense." So all that rambling about the Normans was relevant how? "You don't start counting at a base and work backwards towards the unit." Units are arbitrary, choose a useful one. Mathematics strives for elegance, and being easy to understand and learn is elegant.

alysdexia
Apr 30, 2015
+spaceLem Everything isn't about you.

Usefulness isn't arbitrary but objective. Whereas your constant relies solely on the dimension of r instead of d, with d my constant simplifies in A = 1/2 υd^2, where 1/2 υ shows the areal ratio between a disc inscribed in a square, important as a space-filling parameter.

d as the span is more important than r as the range in many-body and estimation problems. Take the average span and space-filling fraction to find a volume.

Not that my υ relies on d but on rotational transformations or simplices. If the thing were n-ball packing, I'd call on χ = τ/6. More often do you need divisions of τ or π than the latter themselves.
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spaceLem
Apr 30, 2015
+alysdexia I'm sure that comes up all the time, and no doubt it's the first thing you'd want to teach kids who are learning maths, and barely understand circles. r is one dimensional, whether it's a circle, a sphere, a hypersphere, whatever. And since d=2r, so is d.
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alysdexia
May 1, 2015
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+spaceLem Yes, kids understand that width (or breadth or heihth) directly corresponds with span. It's a further concept to take half of a span (range or depth).

spaceLem
May 1, 2015
+alysdexia ...
As for span, it adds ugliness to what should be simple. The radius is the only thing that matters when it comes to circles (unless you're measuring pipes or balls, in which case it's easier to measure the diameter, and then derive the radius, but chances are if you wanted the diameter you're probably done, and won't need to do any further calculations, but for anything else it will be much easier to work with the radius, and that means tau).

alysdexia
May 1, 2015
+spaceLem ...
"for anything else it will be much easier to work with the radius, and that means tau)."

How is it much easier? With the diameter you only need half of the angular sweep. A diameter already bounds a positive measure which you can take two times of.